THE DOUBTEB
less than 2 "with coefficients of the prescribed kind; in fact
1 _- V - 5 is the root of a; - (1 -f V~o) = 0, and the last
coefficient, — (1 -f V — 5), is not a rational integer.
If in the above definition of an algebraic integer of degree »
we suppress the requirement that the leading coefficient be 1,
and say that it can be any rational integer (other than zero,
which is considered an integer), a root of the equation is then
called an algebraic number of degree n. Thus |(1 -f V — 5) is an
algebraic number of degree 2, but is not an algebraic integer; it
is a root of 2x* - 2# + 3 = 0.
Another concept, that of an algebraic number field of degreen, is
now introduced: if r is an algebraic number of degree n, the
totality of all expressions that can be constructed from r by
repeated additions, subtractions, multiplications, and divisions
(division by zero is not defined and hence is not attempted or
permitted), is called the algebraic number field generated by r,
and may be denoted by Ffr], For example, from r we get r + r,
or 2r; from this and r we get 2r/r or 2, 2r — r or r, 2r x r or
2r2, etc. The degree of this F[r] is n.
It can be proved that every member of F[r\ is of the form
Co*"""1 -r C3/n~2 4* - • - T- c«_i? where the c's are rational numbers,
and further every member of F[r] is an algebraic number of
degree not greater than n (in fact the degree is some divisor of
n). Some, but not all, algebraic numbers hi JP[r] will be algebraic
integers.
The central problem of the theory of algebraic numbers is to
investigate the laws of arithmetical divisibility of algebraic
integers in an algebraic number field of degree n. To make this
problem definite it is necessary to lay down exactly what is
meant by 'arithmetical divisibility', and for this we must
understand the like for the rational integers.
"We say that one rational integer, m, is divisible by another, d,
if we canfindarationalinteger,#, such that m — q X d;d (also g)
is called a divisor of m. For example 6 is a divisor of 12, because
12 = 2 x 6; 5 is not a divisor of 12 because there does not exist
a rational integer q such that 12 = q x 5,
A (positive) rational prime is a rational integer greater than 1
MJL—TOL. n. H 519